# Do Stochastic Processes such as the Gaussian Process/Dirichlet Process have densities? If not, how can Bayes rule be applied to them?

The Dirichlet Pocess and Gaussian Process are often referred to as "distributions over functions" or "distributions over distributions". In that case, can I meaningfully talk about the density of a function under a GP? That is, do the Gaussian Process or Dirichlet Process have some notion of a probability density?

If it does not, how can we use Bayes' rule to go from prior to posterior, if the notion of the prior probability of a function is not well defined? Do things such as MAP or EAP estimates exist in the Bayesian Nonparametric world? Thanks a lot.

• Given that the (e.g.) Gaussian process realisation is only observed on a finite collection of points, the corresponding product of the Lebesgue measures is the dominating measure. Which means that for the observation of the random function $f$ at a finite collection of points, there exists a density. Feb 17 '19 at 19:55
• The answer about densities is yes, and the appropriate mathematical formulation is called the Radon-Nikodym derivative.
– whuber
Jul 1 '19 at 21:36

For Gaussian processes there are some cases where we can define a likelihood by using the notion of equivalence of Gaussian measures. An important example is provided by Girsanov's theorem, which is widely used in financial math. This defines the likelihood of an Itô diffusion $$Y_t$$ as the derivative w.r.t the probability distribution of a standard Wiener process $$B_t$$ defined for $$t \geq 0$$. A neat math exposition is found in the book by Bernt Øksendal. The (upcoming) book by Särkkä and Solin provides a more intuitive presentation which will help practitioners. A brilliant math exposition on Analysis and Probability on Infinite-Dimensional Spaces by Nate Elderedge is available.